terrain & typical snow surfaces vary from region to region & resort to resort.
terrain dictates/encourages one pathway versus another.
Here's another factor that influences what teaching path to take. One very very cold day I was working with a day one beginner adult group, all on rental skis. The wax was seriously mis-matched to the conditions du jour and their skis consistently refused to glide.what approach works best for each of his/her individual students.
Our beginner terrain was complicated. The pitch varied. Each part lent itself to learning different skills. The route started out rather steep for them (side-stepping down). This was followed by a gentle and flat pitch (link turns), a chairlift ride (exit the chair safely), then a complicated trail back to the beginning area that alternated from uphill (herringbone to skate,) to super flat (vary turn shape, skate downhill), to a somewhat scary steep (for them) slope where they put it all together.
That day they had to learn to skate. There was no other way to ski downhill given the wax and the snow. By the end of the 2 hour lesson they were all skiing parallel, skating their turns downhill even on the scary pitch, skiing outside foot to outside foot, gliding with each skate. Rudimentary carving was going on. They were laughing.
A full day lesson with first-day beginners that day would have been a hoot, all because of the wax slowing them down.